Residual power of UNF

ABSTRACT

A heat transfer approach to the calculation of residual power of used nuclear fuel (UNF). This application is a conceptual design of an alternative method for determination of residual power of UNF. Our approach is based on the heat transfer analysis of UNF in the transport container with a compact storage cask. To our knowledge, the proposed method for the calculation of residual power of UNF directly in the transport container is unique and can also provide an effective tool to verify the SCALE 6 in order to ensure the safe transport of the UNF.

CROSS-REFERENCES TO RELATED APPLICATIONS

Not Applicable.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

Not Applicable.

MICROFICHE APPENDIX

Not Applicable.

BACKGROUND OF INVENTION

1. Field of Invention

This invention relates to novel alternative methodology to be used forcalculation of residual power of used nuclear fuel (UNF) for the safetransport of radioactive material satisfying the safety standardsrequired by International Atomic Energy Agency.

2. Description of Related Art

The current determination of residual power of nuclear fuel is based onthe SCALE 6 specialized software packages as a tool for WWER-440 fuel.The computations of residual power are performed by the module ORIGEN-S.ORIGEN-S is widely used in nuclear reactor and processing plant designstudies, design studies for used fuel transportation and storage, burnupcredit evaluations, decay heat and radiation safety analyses, andenvironmental assessments.

This module computes the time-dependent concentrations and radiationsource terms of a large number of isotopes, which are simultaneouslygenerated or depleted through neutronic transmutation, fission, andradioactive decay. The computations are based on the system of lineardifferential equations of the first order. The equations describe thecreation and the destruction of nuclides in the fuel. The results ofcomputations are the residual power [W], the activity [Bq], theintensity of photons sources [f/s] and the intensity of neutrons sources[n/s] according to the cooling time.

BRIEF SUMMARY OF INVENTION

Unlike the SCALE 6 system, our approach for the calculation of theresidual power of used nuclear fuel is based on the mathematicalmodeling of heat transfer through the wall of the transport containerwith the used nuclear fuel inside. More precisely, the proposedmethodology is based on measuring the temperature changes of the waterin the container (the values T_(ih,water.in,av)) and the outer walls ofthe container and was applied to the two independent measurements with areal UNF. A direct comparison of the results obtained with the SCALE 6and a combined analytical/experimental heat transfer modeling show goodcorrespondence of the results.

BRIEF DESCRIPTION OF THE TABLES

Having thus described the invention in general terms, reference will nowbe made to the accompanying tables, and wherein:

TAB. 1 contains the values T_(ih,water.in,av), i=0, . . . , N for firstverification experiment.

TAB. 2 contains the values T_(ih,water.in,av), i=0, . . . , N for secondverification experiment.

DETAILED DESCRIPTION OF THE INVENTION

Three simplifications have to be made, to avoid potential problem withderivation of exact mathematical model:

-   -   1) The water temperature T(t) of a homogenized C-30 transport        container with basket KZ-48 depends only on the time and is        spatially uniform within the container.    -   2) The residual power of used nuclear fuel P is constant (i.e.        dP/dt=0) (Remark. Later, we will improve the mathematical model        by an iterative process to enhance the accuracy).    -   3) The container with compact storage cask KZ-48, nuclear fuel        and water is a homogeneous body with specific heat capacity        C_(hom).        The burned-out fuel in a container radiates energy. The radiated        energy is equal to the energy submitted into environment. The        analytical computations for thin-walled vessels adapted to our        situation by homogenization of the container with its content        and using a weighted average show, that the temperature of water        inside the container is governed by the differential equation

$\begin{matrix}{{\frac{{T(t)}}{t} + {\frac{S\; \alpha}{{MC}_{\hom}}\left( {{T(t)} - T_{\infty}} \right)}} = \frac{P}{{MC}_{\hom}}} & (1)\end{matrix}$

with the initial condition

T(0)=T _(0,water.in,av).  (2)

In the above equations are used the following parameters:

-   -   t—the time elapsed from the beginning of measurement i.e.        approximate time since placement in the container [s]    -   C_(hom)—specific heat capacity of the homogenized container C-30        with nuclear fuel basket KZ-48 and nuclear fuel

$\left\lbrack \frac{J}{{{kg}{^\circ}}\mspace{14mu} {C.}} \right\rbrack$

-   -   α—heat transfer coefficient for the surface of a container C-30

$\left\lbrack \frac{W}{m^{2}{^\circ}\mspace{14mu} {C.}} \right\rbrack$

-   -   S—external surface area of the container C-30 [m²]    -   T_(0.water.in,av)—average water temperature detected by the        three sensors located inside the transport container C-30 at the        time t=0 [° C.]    -   T_(∞)—surroundings temperature of the air which are at a        constant (an adjustable air conditioning system was used) [° C.]    -   M—total mass of the C-30 transport container with the nuclear        fuel basket KZ-48 and used nuclear fuel [kg]    -   P—residual power of used nuclear fuel [W].

The unique solution of the initial value problem (1), (2) is thefunction

$\begin{matrix}{{{T(t)} = {\frac{P}{S\; \alpha} + T_{\infty} + {\left( {T_{0,{{water} \cdot {in}},{av}} - \frac{P}{S\; \alpha} - T_{\infty}} \right){\exp \left\lbrack {- \frac{S\; \alpha \; t}{C_{\hom}M}} \right\rbrack}}}},{t \in {\left\lbrack {0,\infty} \right).}}} & (3)\end{matrix}$

Heat is transferred from the water at the higher temperature to the wallof the container with the fins, conducted through the wall, and thenfinally transferred from the cold side of the wall into the surroundingsair at the lower temperature. This series of convective and conductiveheat transfer processes is known as overall heat transfer. In practicegenerally only an average heat transfer coefficient α is required inorder to evaluate the heat power from an area S into the fluid (theair).

For the experiments was used the container C-30 with compact basketKZ-48 with used nuclear fuel. Their parameters are:

-   -   Container C-30 (made of thick-walled structural steel S355JO        (Euronorm))        -   Mass: 67300 [kg]        -   Specific heat capacity:

$C_{C\; 30} = {425 + {0.773T_{{steady},{{coat} \cdot {out}},{av}}} - {0.00169T_{{steady},{{coat} \cdot {out}},{av}}^{2}} + {0.00000222{T_{{steady},{{coat} \cdot {out}},{av}}^{3}\mspace{14mu}\left\lbrack \frac{J}{{{kg}{^\circ}}\mspace{14mu} {C.}} \right\rbrack}}}$

-   -   Compact basket KZ-48 (austenitic stainless steel) with hexagonal        cases (boron steel) containing 48 used nuclear assemblies        -   Mass of basket: 982 [kg]        -   Specific heat capacity:

$C_{{KZ} - 48} = {500\left\lbrack \frac{J}{{{kg}{^\circ}}\mspace{14mu} {C.}} \right\rbrack}$

-   -   -   Mass of cases: 1968 [kg]        -   Specific heat capacity:

$C_{cases} = {475\left\lbrack \frac{J}{{{kg}{^\circ}}\mspace{14mu} {C.}} \right\rbrack}$

-   -   A nuclear fuel assembly comprises a sheath, and nuclear material        (UO₂ tablets) inside the sheath:        -   Cladding material (zirconium alloy)            -   Mass: 4014.48 [kg]            -   Specific heat capacity:

$C_{zircaloy} = {285\left\lbrack \frac{J}{{{kg}{^\circ}}\mspace{14mu} {C.}} \right\rbrack}$

-   -   -   used nuclear fuel—inside the fuel assembly is placed 126            fuel rods about 2.5 [m] long which include ceramic tablets            of uranium dioxide (UO₂ tablets—˜4.8% of actinides)            -   Mass: 6545.52 [kg]            -   Specific heat capacity:

$C_{fuel} = {132.65\left\lbrack \frac{J}{{{kg}{^\circ}}\mspace{14mu} {C.}} \right\rbrack}$

-   -   Total mass (KZ-48 plus fuel rods plus cladding material): 13510        [kg]    -   coolant (water)        -   Mass: 3790 [kg]        -   Specific heat capacity:

$C_{w} \geq {4186\left\lbrack \frac{J}{{{kg}{^\circ}}\mspace{14mu} {C.}} \right\rbrack}$

in the dependence on the temperature.Thus for the homogenized specific heat capacity of the container C-30with nuclear fuel we have

$\begin{matrix}{{C_{\hom} = {\frac{1}{M}\left\lbrack {{67300C_{C\; 30}} + {3790C_{w}} + 3438190.03} \right\rbrack}},{M = 84600}} & (4)\end{matrix}$

where the number

3438190.03=(982×C _(KZ-48))+(1968×C _(cases))+(4014.48×C_(zircaloy))+(6545.52×C_(fuel))=(982×500)+(1968×475)+(4014.48×285)+(6545.52×132.65)

represents a total heat capacity of the basket KZ-48 with the 48 usednuclear assemblies.

For calculating the convection power we obtain, by the limit process fort→∞ in (3),

P=Sα(T(∞)−T _(∞))=Sα(T _(steady,water.in,av) −T _(∞)).  (5)

Due to the idea/strategy of homogenization of the system container pluswater we use for P the modified relation

P=Sα(T _(steady,hom) −T _(∞))  (6)

where

-   -   T_(steady,hom)—mass-weighted steady state temperature of the        container, in our case

$\begin{matrix}{\frac{{T_{{steady},{{water} \cdot {in}},{av}} \times 3790} + {T_{{steady},{{coat} \cdot {out}},{av}} \times 67300}}{3790 + 67300},} & (7)\end{matrix}$

-   -   T_(steady,coat.out,av)—average temperature calculated from the        temperatures detected by sensors (the total amount of them        is 84) situated uniformly on the selected spots on surface of        the container at the time when the average temperature reached a        steady-state [° C.]    -   T_(steady,water.in,av)—average water temperature in the center        of container measured by the three sensors at the time when the        average water temperature reached a steady-state [° C.].        Here we use the concept mass-weighted average temperature i.e.        the mass-weighted average of a quantity is computed by dividing        the summation of the product of density ρ_(i), cell volume, and        the selected field variable (for instance the temperature T_(i))        by the summation of the product of density and cell volume        |V_(i)|

$\frac{\overset{n}{\sum\limits_{i = 1}}{T_{i}\rho_{i}{V_{i}}}}{\sum\limits_{i = 1}^{n}{\rho_{i}{V_{i}}}}.$

The relation (6) will be optimized by an iterative process in theSection below, taking into consideration that the decay heat productionrate will continue to slowly decrease over time.

Only in relatively simple cases, exact values for the heat transfercoefficient α can be found by solving the fundamental partialdifferential equations for the temperature and velocity. An importantmethod for finding the heat transfer coefficients was and still is theexperiment. By measuring the heat flow or flux, as well as the wall andfluid temperatures the local or mean heat transfer coefficient can befound. To completely solve the heat transfer problem all the quantitieswhich influence the heat transfer must be varied when these measurementsare taken. These quantities include the geometric dimensions (e.g.container length and diameter), the characteristic flow velocity and theproperties of the fluid, namely viscosity, density, thermal conductivityand specific heat capacity. To determine the heat transfer coefficientα, we use the mathematical model of heated container (1) and the watertemperature data measured inside the container.

Denote b_(HS)=Sα/C_(hom). Hence

P=C _(hom) b _(HS)(T _(steady,hom) −T _(∞)).  (8)

Thus for relation (3) we have

$\begin{matrix}{{{T(t)} = {\frac{P}{S\; \alpha} + T_{\infty} + {\left( {T_{0,{{{water}.i}\; n},{av}} - \frac{P}{S\; \alpha} - T_{\infty}} \right){\exp \left\lbrack {{- b_{HS}}\frac{t}{M}} \right\rbrack}}}},} & (9)\end{matrix}$

where the coefficient b_(HS) will be calculated by using (9) and theexperimentally obtained data of water heated inside the container byminimizing the distance:

$\begin{matrix}{{\left\lbrack {\sum\limits_{i = 1}^{N}\left( {{T({ih})} - T_{{i\; h},{{{water}.i}\; n},{av}}} \right)^{2}} \right\rbrack \min},} & (10)\end{matrix}$

where

-   -   h is a constant time step of measurement of water temperature,    -   Nh is a time at which a small drop in water temperature is        observed due to the reduction in power of used nuclear fuel in        the container and    -   T_(ih,water.in,av) is a averaged water temperature at the time        ih detected by the sensors located in center of the container        C-30 [° C.].

The Coefficient b_(HS)

Denote {tilde over (h)}=h/M, where h is a time step of measurement (inseconds) and N is a natural number for whichT_(Nh,water.in,av)=T_(steady,water.in,av).

The stationary point and global minimum of (10), the coefficient b_(HS),is an unique solution of transcendental equation

$\begin{matrix}{{{{\sum\limits_{i = 1}^{N}{\; {\exp \left\lbrack {{- 2}\; b_{HS}\overset{\sim}{h}} \right\rbrack}}} = {\sum\limits_{i = 1}^{N}{\; \beta_{i}{\exp \left\lbrack {{- }\; b_{HS}\overset{\sim}{h}} \right\rbrack}}}},{where}}{{\overset{\sim}{h} = \frac{h}{M}},{\beta_{i} = \frac{T_{{steady},{{{water}.i}\; n},{av}} - T_{{ih},{{{water}.i}\; n},{av}}}{T_{{steady},{{{water}.i}\; n},{av}} - T_{{0,{{{water}.i}\; n},{av}}\;}}},{i = 1},{\ldots \mspace{14mu} {N.}}}} & (11)\end{matrix}$

This equation for calculating b_(HS) was obtained as follows. From theequation (5) for steady state regime we have

${T_{{steady},{{{water}.i}\; n},{av}} - T_{\infty}} = \frac{P}{S\; \alpha}$

and thus for (9) we get

${T(t)} = {T_{{steady},{{{water}.i}\; n},{av}} + {\left( {T_{0,{{{water}.i}\; n},{av}} - T_{{steady},{{{water}.i}\; n},{av}}} \right){{\exp \left\lbrack {{- b_{HS}}\; \frac{t}{M}} \right\rbrack}.}}}$

Now differentiating the left side of (10) where

${{T({ih})} = {T_{{steady},{{{water}.i}\; n},{av}} + {\left( {T_{0,{{{water}.i}\; n},{av}} - T_{{steady},{{{water}.i}\; n},{av}}} \right){\exp \left\lbrack {{- b_{HS}}\overset{\sim}{h}} \right\rbrack}}}},{\overset{\sim}{h} = \frac{h}{M}}$

with respect to the variable b_(HS) and equating this to zero we get

${{- 2}{\sum\limits_{i = 1}^{N}{\left\lbrack {T_{{ih},{{{water}.i}\; n},{av}} - T_{{steady},{{{water}.i}\; n},{av}} - {\left( {T_{0,{{{water}.i}\; n},{av}} - T_{{steady},{{{water}.i}\; n},{av}}} \right){\exp \left\lbrack {{- b_{HS}}\overset{\sim}{h}} \right\rbrack}}} \right\rbrack \left( {T_{{steady},{{{water}.i}\; n},{av}} - T_{0,{{{water}.i}\; n},{av}}} \right)\overset{\sim}{h}\; {\exp \left\lbrack {{- b_{HS}}\overset{\sim}{h}} \right\rbrack}}}} = 0.$

Hence

${\sum\limits_{i = 1}^{N}{\left\lbrack {T_{{ih},{{{water}.i}\; n},{av}} - T_{{steady},{{{water}.i}\; n},{av}} - {\left( {T_{0,{{{water}.i}\; n},{av}} - T_{{steady},{{{water}.i}\; n},{av}}} \right){\exp \left\lbrack {{- b_{HS}}i\overset{\sim}{h}} \right\rbrack}}} \right\rbrack \; {\exp \left\lbrack {{- b_{HS}}\overset{\sim}{h}} \right\rbrack}}} = 0$

and finally, after simple algebraic manipulation we have (11).

Coefficient b_(HS) Optimization Algorithm

We use the b_(HS) coefficient-optimizing algorithm taking intoconsideration reduction in power of used nuclear fuel in the container.

We apply the following iterative scheme:

$\begin{matrix}{{{{\sum\limits_{i = 1}^{N}{\; {\exp \left\lbrack {{- 2}\; b_{HS}^{(k)}\overset{\sim}{h}} \right\rbrack}}} = {\sum\limits_{i = 1}^{N}{\; \beta_{i}^{(k)}{\exp \left\lbrack {{- }\; b_{HS}^{(k)}\overset{\sim}{h}} \right\rbrack}}}},{where}}{{\beta_{i}^{(k)} = \frac{\left( {T_{{steady},{{{water}.i}\; n},{av}} + \Delta^{({k - 1})}} \right) - T_{{ih},{{{water}.i}\; n},{av}}}{\left( {T_{{steady},{{{water}.i}\; n},,{av}} + \Delta^{({k - 1})}} \right) - T_{0,{{{water}.i}\; n},{av}}}},{i = 1},{\ldots \mspace{14mu} N},{\Delta^{(0)} = 0},{\Delta^{(k)} = \frac{\left( {T_{{steady},{{{water}.i}\; n},{av}} - T_{0,{{{water}.i}\; n},{av}}} \right){\exp \left\lbrack {{- b_{HS}^{(k)}}N\overset{\sim}{h}} \right\rbrack}}{1 - {\exp \left\lbrack {{- b_{HS}^{(k)}}N\overset{\sim}{h}} \right\rbrack}}},{k = 1},\ldots}} & (12)\end{matrix}$

To solve the equation (12) we use the mathematical software package.

An iterative process is finished when a stopping criterion is achieved,

|b _(HS) ^((k)) −b _(HS) ^((k−1))|≦ε(ε=10⁻³, for example).

Thus, the optimized relation (8) for residual power of used nuclear fuelis

P=C _(hom) b _(HS) ^((k))(T _(steady,hom)+Δ^((k−1)) −T _(∞)).  (13)

Quality of Optimization of b_(HS) ^((k))

The quality of optimization Λ^((k)) of the coefficient b_(HS) ^((k))(i.e. of k-th iteration) can be determined from the relation

$\begin{matrix}{{{\Lambda^{(k)} = {{T_{{Nh},{{{water}.i}\; n},{av}} - {T^{(k)}({Nh})}} = {\left\{ {\left( {T_{{steady},{{{water}.i}\; n},{av}} + \Delta^{({k - 1})} - T_{0,{{{water}.i}\; n},{av}}} \right){\exp \left\lbrack {{- b_{HS}^{(k)}}N\overset{\sim}{h}} \right\rbrack}} \right\} - \Delta^{({k - 1})}}}},\mspace{20mu} {where}}{{{T^{(k)}(t)} = {T_{{steady},{{{water}.i}\; n},{av}} + \Delta^{({k - 1})} + {\left( {T_{0,{{{water}.i}\; n},{av}} - T_{{steady},{{{water}.i}\; n},{av}} - \Delta^{({k - 1})}} \right){\exp \left\lbrack {{- b_{HS}^{(k)}}\frac{t}{M}} \right\rbrack}}}},}} & \; \\{\mspace{20mu} {{k = 1},2,\ldots}} & \;\end{matrix}$

Hence

T ^((k))(∞)=T _(steady,water.in,av)+Δ^((k−1)) ,k=1,2, . . .

A smaller value of Λ^((k)) implies a more accurate approximation ofb_(HS).

Now the proposed method will be illustrated and validated by using thereal data.

Proposed Method Application for the Container with UNF.

The First Experiment

Residual power calculated by the SCALE 6 system: P_(SCALE6)=17309 [W].

The input data are the following:

-   -   h=7200    -   {tilde over (h)}=7200/84600    -   N=40    -   T_(0,water.in,av)=63.3    -   T_(∞)=21    -   T_(water.in,av)=71        C_(w)=4193    -   T_(steady,water.in,av)=73.8    -   T_(steady,coat.out,av)=50.34.

The Table I contains the key values for determining the coefficientsb_(HS) ^((k)) and Δ^((k)) for (13).

Using the iterative scheme as is presented in the Section, we obtain

-   -   b_(HS) ⁽¹⁾=1.051, Δ⁽¹⁾=0.301, Λ⁽¹⁾=0.293    -   b_(HS) ⁽²⁾=0.983, Δ⁽²⁾=0.382, Λ⁽²⁾=0.078    -   b_(HS) ⁽³⁾=0.966, Δ⁽³⁾=0.407, Λ⁽³⁾=0.023    -   b_(HS) ⁽⁴⁾=0.960, Δ⁽⁴⁾=0.414, Λ⁽⁴⁾=0.007    -   b_(HS) ⁽⁵⁾=0.959, Δ⁽⁵⁾=0.417, Λ⁽⁵⁾=1.452×10⁻¹⁰        and from (4) we get C_(hom)≐594.35.

Further, from (7) we have

$T_{{steady},\hom} = {\frac{{73.8 \times 3790} + {50.34 \times 67300}}{3790 + 67300} = {{51.591\left\lbrack {{^\circ}\mspace{14mu} {C.}} \right\rbrack}.}}$

Substituting these values into (13) (with k=5) we obtain

P=C _(hom) b _(HS) ⁽⁵⁾(T _(steady,hom)+Δ⁽⁴⁾ −T_(∞))=594.35×0.959×(51.591+0.414−21)≐17672[W].

Proposed Method Application for the Container with UNF.

The Second Experiment

Residual power calculated by the SCALE 6 system: P_(SCALE6)=16355 [W].

The input data are the following:

-   -   h=7200    -   {tilde over (h)}=7200/84600    -   N=59    -   T_(0,water.in,av)=44.7    -   T_(∞)=18    -   T_(water.in,av)=65        C_(w)=4190    -   T_(steady,water.in,av)=71.8    -   T_(steady,coat.out,av)=50.26.

In the Table II are the values of water temperature for determining thecoefficients b_(HS) ^((k)) and Δ^((k)) for (13).

Using these values for the iterative procedure we get

-   -   b_(HS) ⁽¹⁾=0.826, Δ⁽¹⁾=0.434, Λ⁽¹⁾=0.427    -   b_(HS) ⁽²⁾=0.793, Δ⁽²⁾=0.514, Λ⁽²⁾=−5.273×10⁻¹¹    -   b_(HS) ⁽³⁾=0.786, Δ⁽³⁾=0.534, Λ⁽³⁾=−1.099×10⁻¹⁰    -   b_(HS) ⁽⁴⁾=0.784, Δ⁽⁴⁾=0.538, Λ⁽⁴⁾=4.102×10⁻¹⁰.

Similarly as for the first measurement we obtain C_(hom)≐594.17 and from(7) we have

$T_{{steady},\hom} = {\frac{{71.8 \times 3790} + {50.26 \times 67300}}{3790 - 67300} = {{51.4083556\;\left\lbrack {{^\circ}\mspace{14mu} {C.}} \right\rbrack}.}}$

Substituting these values into (13) (with k=4) we obtain

P=(C _(hom) b _(HS) ⁽⁴⁾(T _(steady,hom)+Δ⁽³⁾ −T_(∞))=594.17×0.784×(51.41+0.534−18)=15812[W].

CONCLUSIONS

The proposed method application leads to the good results. Thepercentage difference between the results achieved by the SCALE 6 systemand our method based on the heat transfer analysis is

${{\frac{P_{{SCALE}\; 6} - P}{P_{{SCALE}\; 6}} \cdot 100}\%} = {{{\frac{17309 - 17672}{17309} \cdot 100}\%} = {{- 2.09}\%}}$

for the first measurement and

${{\frac{P_{{SCALE}\; 6} - P}{P_{{SCALE}\; 6}} \cdot 100}\%} = {{{\frac{16355 - 15812}{16355} \cdot 100}\%} = {3.3\%}}$

for the second one, which is negligible for this type of calculation andis within the range of measurement uncertainty (3-5%). Since an exactmathematical modeling of the thermal processes in the system containerplus water plus used nuclear fuel with non-uniform burnup distributionsis impossible, inter alia some of the parameters of the model may bedetermined experimentally only (for instance a heat transfer coefficientα).

TABLE I The values T_(ih, water. in, av), i = 0, . . . , N. Firstexperiment i 0 1 2 3 4 5 6 7 8 T_(ih, water. in, av) 63.3 64.2 64.9 65.666.2 66.7 67.2 67.7 68.1 i 9 10 11 12 13 14 15 16 17T_(ih, water. in, av) 68.5 68.9 69.4 69.8 70.2 70.6 71.0 71.3 71.6 i 1819 20 21 22 23 24 25 26 T_(ih, water. in, av) 71.9 72.1 72.4 72.6 72.873.0 73.1 73.2 73.2 i 27 28 29 30 31 32 33 34 35 T_(ih, water. in, av)73.3 73.3 73.3 73.4 73.5 73.5 73.6 73.6 73.6 i 36 37 38 39 40T_(ih, water. in, av) 73.7 73.7 73.7 73.7 73.8

TABLE II The values T_(ih, water. in, av), i = 0, . . . , N. Secondexperiment i 0 1 2 3 4 5 6 7 8 T_(ih, water. in, av) 44.7 46.8 48.7 50.451.9 53.4 54.7 55.9 57.0 i 9 10 11 12 13 14 15 16 17T_(ih, water. in, av) 58.1 58.9 59.9 60.7 61.5 62.1 62.8 63.4 63.9 i 1819 20 21 22 23 24 25 26 T_(ih, water. in, av) 64.4 64.9 65.3 65.7 66.166.4 66.7 67.0 67.3 i 27 28 29 30 31 32 33 34 35 T_(ih, water. in, av)67.5 67.7 68.0 68.1 68.3 68.5 68.6 68.8 68.9 i 36 37 38 39 40 41 42 4344 T_(ih, water. in, av) 69.0 69.1 69.2 69.3 69.4 69.5 69.6 69.7 69.7 i45 46 47 48 49 50 51 52 53 T_(ih, water. in, av) 69.8 70.0 70.1 70.370.5 70.7 70.8 71.0 71.1 i 54 55 56 57 58 59 T_(ih, water. in, av) 71.271.4 71.5 71.6 71.7 71.8

Having described our invention, we claim:
 1. a heat transfer approach tothe calculation of residual power of used nuclear fuel comprising a. theidea of examining the relationship between the temperature dynamics ofthe water in the container and the temperature of outer walls of thecontainer and residual power of used nuclear fuel; b. a mathematicalmodel (1) for heat transfer through the wall of homogenized transportcontainer with used nuclear fuel inside; c. an analytic/experimentalmethod for determination of coefficient b_(HS)=Sα/C_(hom).
 2. a heattransfer approach to the calculation of residual power of used nuclearfuel as recited in claim 1, further comprising the application to alltypes of transport containers intended for used nuclear fuel.